Sampling and Cubature on Sparse Grids Based on a B-spline Quasi-Interpolation

Abstract

Let $$X_n = \{x^j\}_{j=1}^n$$Xn={xj}j=1n be a set of n points in the d-cube $${\mathbb {I}}^d:=[0,1]^d$$Id:=[0,1]d, and $$\Phi _n = \{\varphi _j\}_{j =1}^n$$źn={źj}j=1n a family of n functions on $${\mathbb {I}}^d$$Id. We consider the approximate recovery of functions f on $${{\mathbb {I}}}^d$$Id from the sampled values $$f(x^1), \ldots , f(x^n)$$f(x1),ź,f(xn), by the linear sampling algorithm $$ L_n(X_n,\Phi _n,f) := \sum _{j=1}^n f(x^j)\varphi _j. $$Ln(Xn,źn,f):=źj=1nf(xj)źj. The error of sampling recovery is measured in the norm of the space $$L_q({\mathbb {I}}^d)$$Lq(Id)-norm or the energy quasi-norm of the isotropic Sobolev space $$W^\gamma _q({\mathbb {I}}^d)$$Wqź(Id) for $$1 0. Functions f to be recovered are from the unit ball in Besov-type spaces of an anisotropic smoothness, in particular, spaces $$B^{\alpha ,\beta }_{p,\theta }$$Bp,źź,β of a hybrid of mixed smoothness $$\alpha > 0$$ź>0 and isotropic smoothness $$\beta \in {\mathbb {R}}$$βźR, and spaces $$B^a_{p,\theta }$$Bp,źa of a nonuniform mixed smoothness $$a \in {\mathbb {R}}^d_+$$aźR+d. We constructed asymptotically optimal linear sampling algorithms $$L_n(X_n^*,\Phi _n^*,\cdot )$$Ln(Xnź,źnź,·) on special sparse grids $$X_n^*$$Xnź and a family $$\Phi _n^*$$źnź of linear combinations of integer or half integer translated dilations of tensor products of B-splines. We computed the asymptotic order of the error of the optimal recovery. This construction is based on B-spline quasi-interpolation representations of functions in $$B^{\alpha ,\beta }_{p,\theta }$$Bp,źź,β and $$B^a_{p,\theta }$$Bp,źa. As consequences, we obtained the asymptotic order of optimal cubature formulas for numerical integration of functions from the unit ball of these Besov-type spaces.